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\centerline{\cmrX  About the Unduloid } 
\smallskip
%\centerline{\cmrX }
%\centerline{H. Karcher, Version Oct. 2004}

\noindent
What are the different shapes that a soap film can take, or to put it 
somewhat differently, what can we say about the differential geometry 
of a mathematical surface that approximates a soap film? 


\noindent
An important  physical characteristic of the soap film is its surface 
tension $T$. This depends only on the chemical composition of the
liquid from which it is made, and so it is the same at each point of the
film. The difference in air pressure between the two sides of the film 
is an environmental variable that is also clearly the same at all points 
of the film. Now it follows from physical principles (that we will take for 
granted here) that the mean curvature $H$ of the soap film at any 
point is equal to  $P/T$, and so we see that a soap film is always 
represented by a surface that has constant mean curvature.

\noindent
For a soap film that we get by dipping a closed loop of wire 
into soapy water, the air pressure on both sides is clearly the same, so such 
a soap film must have mean curvature zero. Such surfaces are called 
{\it minimal surfaces\/}, since it can be shown that if we draw any small closed 
curve on the surface, the area of the part of the surface inside the curve is less
than or equal to the area of any other surface bounded by the curve.

\noindent
We consider minimal surfaces in considerable detail elsewhere, and here 
we shall be interested in the case of soap bubbles. These are soap films that
(perhaps together with some other surfaces) enclose a bounded region of 
space (the ``inside'' of the bubble). For bubbles the pressure will be slightly 
greater on the inside than on the outside, so that the surface is what is called 
a {\it CMC surface\/}, that is it has {\bf non-zero} constant mean curvature (and 
of course for the floating type it is often just a sphere). 

  
 \noindent                     
 If one blows a soap bubble between two parallel glass plates then one 
can obtain CMC surfaces that are surfaces of revolution, and such 
CMC surfaces are called {\it Unduloids\/}.
 
  \noindent 
Consider a curve in the $x$-$y$-plane, given parametrically by 
$x=x(t), y=y(t)$, or as a graph $(x,f(x))$ of a function $f$.  
If one rotates this curve about the
$x$-axis, it is easy to compute an expression for the mean curvature $H$
of the resulting surface of revolution in terms of the first and second
derivative of $x(t)$ and $y(t)$  (or, in the graph description, the derivatives 
of $f$). If this expression is set equal to a positive constant $H$, 
one gets differential equations for the functions $x(t)$ and $y(t)$ 
(respectively for the function $f$), and solving these ODE provides a 
method for finding 
all CMC surfaces of revolution. Delaunay studied this problem in 1841, 
and being an expert on the theory of roulettes (i.e., a locus traced out 
by a point attached to curve as that curve rolls on a line), he recognized 
that the solutions of this differential equation could be identified with
the roulettes traced out by a focus of a conic section as it rolls along
the $x$-axis. The special case that the conic is an ellipse gives the
Unduloid. In 3D-XplorMath, the Unduloid is literally constructed
by this double process of first rolling an ellipse and tracking one of
its foci and then rotating the resulting curve around the $x$-axis.
  
  \noindent
The default morph shows a family of unduloids that starts with
a cylinder and deforms towards a chain of spheres. With the
rolling construction of the Unduloid, we cannot reach the chain
of spheres because the parameter lines become concentrated near
the narrowing necks of the surfaces.  However, if one resizes these 
necks so they have constant waist size, then the necks converge
to (minimal) Catenoids. This fact was very important in the
construction of very general examples by Kapouleas.
  
\noindent
 H.K.



\bye
